Random Variable

Definition:
A random variable
is defined as a real- or complex-valued
function of some random event, and is fully characterized by its
probability distribution.

Example:
A random variable can be defined based on a coin toss by defining
numerical values for heads and tails. For example, we may assign 0 to
tails and 1 to heads. The probability distribution for this random
variable is then

(C.4)

Example:
A die can be used to generate integer-valued random variables
between 1 and 6. Rolling the die provides an underlying random event.
The probability distribution of a fair die is the
discrete uniform distribution between 1 and 6. I.e.,

(C.5)

Example:
A pair of dice can be used to generate integer-valued random
variables between 2 and 12. Rolling the dice provides an underlying
random event. The probability distribution of two fair dice is given by

(C.6)

This may be called a discrete triangular distribution. It can
be shown to be given by the convolution of the discrete uniform
distribution for one die with itself. This is a general fact for sums
of random variables (the distribution of the sum equals the
convolution of the component distributions).

Example:
Consider a random experiment in which a sewing needle is dropped onto
the ground from a high altitude. For each such event, the angle of
the needle with respect to north is measured. A reasonable model for
the distribution of angles (neglecting the earth's magnetic field) is
the continuous uniform distribution on
, i.e., for
any real numbers
and
in the interval
, with
, the probability of the needle angle falling within that interval
is

(C.7)

Note, however, that the probability of any single angle
is zero. This is our first example of a
continuous probability distribution.
Therefore, we cannot simply define the
probability of outcome
for each
.
Instead, we must define the probability density function
(PDF):

(C.8)

To calculate a probability, the PDF must be integrated over one or more
intervals. As follows from Lebesgue integration theory (``measure theory''),
the probability of any countably infinite set of discrete points is
zero when the PDF is finite. This is because such a set of points is
a ``set of measure zero'' under integration. Note that we write
for discrete probability distributions and
for PDFs. A discrete probability distribution such as that in
(C.4) can be written as